Algebraic {$q$}-Integration and Fourier Theory on Quantum and Braided Spaces

TL;DR

This paper develops an algebraic framework for integration and Fourier analysis on quantum and braided spaces, revealing new interpretations of $q$-integrals and establishing a $q$-Fourier transform with convolution properties.

Contribution

It introduces an algebraic approach to $q$-integration and Fourier theory on braided spaces, including quantum planes, with novel algebraic interpretations and generalizations.

Findings

Algebraic interpretation of Jackson $q$-integral as indefinite integration on braided groups

Development of a $q$-Fourier transform with convolution theorem and $F^2=S$ property

Extension of the theory to quantum spaces associated with general R-matrices

Abstract

We introduce an algebraic theory of integration on quantum planes and other braided spaces. In the one dimensional case we obtain a novel picture of the Jackson $q$-integral as indefinite integration on the braided group of functions in one variable $x$. Here $x$ is treated with braid statistics $q$ rather than the usual bosonic or Grassmann ones. We show that the definite integral $\int x$ can also be evaluated algebraically as multiples of the integral of a $q$-Gaussian, with $x$ remaining as a bosonic scaling variable associated with the $q$-deformation. Further composing our algebraic integration with a representation then leads to ordinary numbers for the integral. We also use our integration to develop a full theory of $q$-Fourier transformation $F$. We use the braided addition $\Delta x=x\otimes 1+1\otimes x$ and braided-antipode $S$ to define a convolution product, and prove a convolution theorem. We prove also that $F^2=S$. We prove the analogous results on any braided group, including integration and Fourier transformation on quantum planes associated to general R-matrices, including $q$-Euclidean and $q$-Minkowski spaces.